Integrability Aspects of the Current Algebra Representation and the Factorized Quantum Nonlinear Schrodinger Type Dynamical Systems

In this work we study integrability aspects of the current algebra representation and the factorized quantum nonlinear Schrodinger type dynamical systems, initiated before by G. Goldin with collaborators, in suitably renormalized Fock type Hilbert spaces. There is developed the current algebra representation scheme of reconstructing algebraically factorized quantum Hamiltonian and symmetry operators in the Fock type space. There is presented its application to constructing quantum factorized Hamiltonian systems and their symmetry operators in case of quantum integrable spatially many- and one-dimensional dynamical systems. As examples we have studied in detail the factorized structure of Hamiltonian operators, describing such quantum integrable spatially one-dimensional models as the Calogero-Moser-Sutherland and Nonlinear Schrodinger dynamical systems of spin-less Bose-particles.